On the non-asymptotic concentration of heteroskedastic Wishart-type matrix

Abstract

This paper focuses on the non-asymptotic concentration of the heteroskedastic Wishart-type matrices. Suppose Z is a p1-by-p2random matrix and Zij∼N(0,σij2) independently, we prove the expected spectral norm of Wishart matrix deviations (i.e., EZZ⊤−EZZ⊤) is upper bounded by (1+ϵ)2σCσR+σC2+Cσ Rσ∗ log(p1∧p2)+Cσ∗2log(p 1∧p2), where σC2:=maxj ∑i=1p1σ ij2, σR2:=maxi ∑j=1p2σ ij2 and σ∗2:=maxi,jσij2. A minimax lower bound is developed that matches this upper bound. Then, we derive the concentration inequalities, moments, and tail bounds for the heteroskedastic Wishart-type matrix under more general distributions, such as sub-Gaussian and heavy-tailed distributions. Next, we consider the cases where Z has homoskedastic columns or rows (i.e., σij≈σi or σij≈σj) and derive the rate-optimal Wishart-type concentration bounds. Finally, we apply the developed tools to identify the sharp signal-to-noise ratio threshold for consistent clustering in the heteroskedastic clustering problem.